Axiomatization of stochastic models for choice under uncertainty

This paper develops a theory of probabilistic models for risky choices. This theory can be viewed as an extension of the expected utility theory. One probabilistic version of the Archimedean Axiom and two versions of the Independence Axiom are proposed. In addition, additional axioms are proposed of which one is Luce's Independence from Irrelevant Alternatives (IIA). It is demonstrated that different combinations of the axioms yield different characterizations of the probabilities for choosing the respective risky prospects. Particular dimensional invariance axioms are postulated for the case with monetary rewards. It is demonstrated that when probabilistic versions of the Archimedean and the Independence Axioms are combined with Dimensional Invariance axioms explicit functional forms of the utility function follow. It is also proved that a random utility representation exists in the particular case when IIA holds for choice among lotteries. An interesting feature of the models developed is that they allow for violations of the expected utility theory known as the common consequence effect and the common ratio effect.